Parametric Amplification of Scattered Atom Pairs

NonLinear Atom Optics

Using the strong nonlinearity provided by s-wave colllisons, a number of nonlinear processes first observed with photons have been observed with matter waves. Nonlinear atom optics, and in particular four-wave mixing has been suggested as an ideal way to create entangled pairs of atoms. However, in previous experiments [1,2] due to the quadratic dispersion relation for free particles, the phase-matching condition was only fulfilled when the magnitudes of the momenta (in the moving frame) were equal.

Recently [3], it was suggested that if a one dimensional lattice is added, the resulting band structure would allow for non-degenerate four wave mixing in one dimension, and if seeded the process should be amplified.

Collistions in Moving Condensates

In free space, if a condensate is moving with momentum k0, atoms within the condensate cannot elastically scatter into different momentum states due to the quadratic dispersion relation.

However, if a 1D lattice is added this is no longer true. Even for small lattice depths, the dispersion relation is no longer quadratic near the boundary of the first Brillouin zone, allowing elastic scattering to occur for some values of k0

For different values of k0, the allowed quasimomentum states, k1 and k2, also vary. For values of k0 below ~.55 kL, where kL is the wavector of the optical lattice, elastic scattering cannot occur.

Fig. 1. Band structure for a lattice depth of V=.5Erec. The
dashed line shows the free particle dispersion curve. The dispersion
relation of the lattice allows two atoms with momentum k0 to elastically
scatter into the final momentum states k1 and k2. Energy and
quasimomentum are conserved when k0 is the average of k1 and k2
and the three points on the dispersion curve lie on a straight line. If
k0 is varied, the allowed values for k1 and k2 change. For values of
k0 below 0.55kL, where kL is the wavevector of the optical lattice,
atoms cannot scatter elastically into different momentum states.
The circles (squares) show allowed states k0,k1,k2 for k0 =0.66kL
(0.70kL). As k0 is increased, the final momentum states move closer
together. Since the scattering occurs within the lowest band of the lattice,
the final momentum is k2 = (2k0 - k1)Mod(2kL)

Parametric Generation

A 87Rb Bose-Einstein condensate with quasimomentum k0 was
loaded into a one-dimensional optical lattice. To load the
atoms at a given quasimomentum relative to the Brillouin
zone, a moving optical lattice was adiabatically applied to
a magnetically trapped condensate initially at rest in the lab
frame. The lattice was created using two counter-propagating
laser beams with frequency difference dn, giving the lattice a
velocity of v =l/2 dn, where l is the wavelength of the optical
lattice. In the rest frame of the lattice, the condensate has
quasimomentum k0 = ml/2hbar dn, where m is the atomic mass.
By changing the detuning between the lattice beams, dn, the
quasimomentum k0 of the condensate could be varied. As
shown in Fig. 2, as the value of k0 was varied we observed
elastic scattering of atom pairs into final states k1 and k2.

This process is analogous to optical parametric generation (OPG) with photons

Fig. 2.Experimental Setup for Parametric Generation. (a)A 87 Rb
Bose-Einstein condensate is illuminated by two counter-propagating
laser beams with detuning dn, which create a moving optical lattice.
The condensate is initially held at rest in a magnetic trap. (b) As the quasimomentum k0 of the condensate was varied, we observed elastic scattering into states k1 and k2. (c) Absorption images for different lattice detunings, dn showing parametric
generation. After ramping up the lattice, the atoms were held for
10 ms at a constant lattice depth. They were then released from the
trap and imaged after 43 ms of ballistic expansion. The field of view
is 0.5 mm×0.3 mm.

Parametric Amplification

To demonstrate the matter wave analogue of a optical parametric amplification (OPA), we first created a small seed of atoms with quasimomentum k1 before ramping on the moving lattice (see Fig. 3). To create the seed we applied a Bragg pulse to the magnetically trapped condensate, outcoupling a small fraction of atoms into the momentum state kBragg. Immediately after applying the pulse, the optical lattice was adiabatically ramped on. In the rest frame of the lattice, the seed has quasimomentum k1 = kBragg + k0. The phase matching condition for a given seed momentum kBragg was found by varying the frequency difference dn of the lattice, and therefore the quasimomenta k0 and k1 of the atoms. As shown in Fig. 3, when the phase matching condition was fulfilled, we observed amplification of the seed k1 as well as its conjugate momentum k2.

Fig. 3. Parametric amplification of scattered atom pairs in a 1D optical
lattice. (a) First, a 2 ms Bragg pulse was applied to the condensate.
(b) The Bragg pulse seeded atoms along the long axis of
the condensate with momentum kBragg = (ka -kb) in the lab frame
(c) The optical lattice was then adiabatically ramped on and applied
for 10 ms. When the phase matching condition was fulfilled, parametric
amplification of atoms in the seeded state k1 and its conjugate
momentum state k2 was observed. (d) Absorption images showing amplification of
k1 and k2 when the phase matching condition is met. The center
of the resonance was at dn = 5450Hz, close to the calculated value
of dn = 5350Hz. The width of the resonance is determined by the
Fourier width of the Bragg pulse. Most of the scattered atoms in the
third image were independent of the seed pulse.

1. L. Deng et .al, Nature 398, 218(1999).
2. J. M. Vogels et. al, Phys. Rev. Lett 89, 020401,(2002).
3. K.M. Hilligsøe, and K.Mølmer, Phys. Rev. A 71, 041602(R) (2005).
4. Gretchen K. Campbell, Jonchul Mun, Micah Boyd, Erik W. Streed, Wolfgang Ketterle and David E. Pritchard: Parametric Amplification of Scattered Atom Pairs.
   Accepted for publication in Physical Review Letters (2005)
   Preprint